+ O() → log g2N 8π − γ
(4.111) I(g2, Nf = 0) = log
−g2N 8π − γ
= ˜I(g2) ± iπ. (4.112)
Therefore, for Nf = 0, the bion amplitude has an ambiguity, [Bii]θ± = [KiK¯i]θ± = (log g2N
8π
− γ)2A2ie−2Sk± iπ2A2ie−2Sk, (4.113)
where θ± denotes the direction we chose to do analytic continuation, Sk = SNI is the action of the kink-instanton and γ is the Euler-Mascheroni constant. So there is a non-perturbative ambiguous imaginary part arising from the bion confiugration in the pure bosonic theory, whereas for Nf ≥ 1, there is no ambiguity in the bion configuration.
Let’s focus on the next leading topological molecules, the bion-anti-bion molecules [BijBji]. The quasi-zero mode integrals is given by,
I(g2) = Z ∞
0
dτ exp(V (τ )), V (τ ) = (µB, µB)8ζ
g2e−ζτ, (4.114) where µB := αi− αj ∈ Γ∨r. Again, this integral is dominated by small τ and we need to do regularization. It gives us an amplitude of the form,
[BijB¯ij]θ± = Re[BijB¯ij] + i Im[BijB¯ij]θ± ∼ e−4Sk ± iπe−4Sk. (4.115) This is the leading ambiguity in the theory with Nf ≥ 1 fermions.
4.2.5 IR renormalons and bions
We have seen that when QCD(adj) is compactified in a paticular way, there are new semiclassical configurations in it. Those new semiclassical configurations (monopole-instantons) form new topological molecules (bions, bion-anti-bion pairs). Some of the topological molecules ([BijBji], [BijBjkBki]) have ambiguous imaginary parts, just like the instanton-anti-instanton pair in QM. We have conjectured that these
7Actually, the quasi-zero mode integral is divergent. The divergent term is due to the large separation of the kink and anti-kink, the double-counting of the uncorrelated [K] − [ ¯K] events. It should be subtracted off.
imaginary parts can cancel the imaginary parts of the IR renormalons in non-compactified QCD(adj). However, the order of the imaginary part of the IR renor-malon is not the same as the topological molecules, so it is impossible for them to cancel each other.
In CPN −1model, by similar procedure, we found new semiclassical configurations (kinks). These kinks can form bions ([Bii] = KiK¯i) and the combinations of the bions have ambiguous imaginary parts. They give us imaginary parts of order e−SIN and is of the same order as the imaginary parts of the IR renormalons. Therefore, the ambiguity we have found in compactified theory may cancel the ambiguity of the IR renormalons in non-compact theory.
Let’s start from the perturbation series in CPN −1 model. We denote asymptotic perturbative series in CPN −1 model by P (g2),
P (g2) =
∞
X
n=0
ang2n. (4.116)
We define the Borel transform of P (g2) by BP(g2),
BP(g2) :=
∞
X
n=0
an
n!g2n. (4.117)
And we define the Borel resummation of P (g2) by B(g2)8,
B(g2) = Z ∞
0
e−tBP(g2t)dt (4.118) If the function BP(g2t) has no pole on the positive real axis, then this integration can be done and B(g2) is real. If there are poles lying on the positive real axis, the series is non-Borel summable. If we still want to define the sum, we need to change the integration contour to avoid the poles. The integration has ambiguities, this is called the lateral Borel sum. The ambiguities depend on whether the choice of the integration path is above or below the poles,
B(g2)θ±= Re B(g2) ± i Im B(g2). (4.119)
8We often define the divergent series P (g2) as its Borel resummation B(g2), so P (g2) and B(g2) are the same thing
Figure 9: Lateral Borel sums, the different choices of the path give us different results.
In CPN −1 model on R2, the leading imaginary part of the Borel sum is of the form [36, 37],
Im B(g2) ∼ e−2nSIβ0 ∼ πe−2nSIN , n = 2, 3, . . . , (4.120) which comes from the IR renormalons. Let’s recall the semiclasscial configurations we have found in CPN −1 on R × S1; we have N types of kinks Ki, i = 1, 2 . . . N . A kink-anti-kink pair forms a bion,
KiK¯i = [Bii], (4.121)
For Nf ≥ 1, since kink configuration carries fermionic zero modes, the bion ampli-tude is real and it has no ambiguity. The leading ambiguity appears at the 4-th order, the bion-anti-bion amplitude. The ambiguity of the bion-anti-bion amplitude is of order e−4SIN , and the leading IR renormalon is also of order e−4SIN . So if the imaginary part of the bion-anti-bion in the compactified theory and the imaginary part of the IR renormalon in the original theory cancels each other,
Im B(g2)θ± on R2+ Im[BijB¯ij]θ± on R1× S1 = 0, to order of e−4SIN , (4.122) then there must be a deep relation between the CPN −1on R2 and CPN −1on R1×S1 with periodic boundary condition on fermions.
For Nf = 0, there is an ambiguous imaginary part in bion amplitudes. Therefore, in a pure bosonic CPN −1 theory, the leading ambiguity is of order e−2SIN , but the closest IR renormalon on R2 is of order e−4SIN so there is a mismatch. There are
-6
CPN −1 on R2 t
t t
t t t t d t
UV renormalons
?
t = −8πn/g2β0
IR renormalons
?
t = 8πn/g2β0 n=2,3,. . .
Instanton-anti-instanton pole t = 8π/g2, 16π/g2, . . .
?
-6
CPN −1 on R1× S1 t
t t
t t t t d t
UV renormalons
?
t = −8πn/g2N
IR renormalons
?
t = 8πn/g2N n=19,2,. . .
Instanton-anti-instanton pole t = 8π/g2, 16π/g2, . . .
?
Figure 10: Upper figure: The conjectured pole structure of the Borel plane for CPN −1 on R2. Lower figure: The semiclassical singularities corresponding to the topological molecules in CPN −1on small R×S1. There is an extra singularity closer to the origin for Nf = 0. For Nf ≥ 1, the pole structure of the topological molecules and the IR renormalons concide. Although the analysis is done in weakly coupled region and the IR renormalons come from strongly coupled region, we conjectured that the pole structure does not change significantly. This is an evidence to show that bion-anti-bion pairs on R1 × S1 may be the semiclassical realization of the IR renormalons on R2.
several possibilities. First possibility is that the IR renormalons can not always be realized by semiclassical configurations. Namely, there would be two types of singularities on the Borel plane, one type is the instanton pole which can be realized semiclassically, the other type is the IR renormalon which can not be realized by the semiclassical configurations. Second possibility is that the conjectured Borel plane structure for Nf = 0 CPN −1model is wrong, there is also an extra singularity appearing closer to the origin when Nf = 0, but we have not found this kind of thing happen yet. Third, some kind of phase transition may have happend when we change the length of the compactified direction from small-L limit to large-L, the analysis we have done in weak coupling region does not work for strong coupling.
Let’s see the cancellation to the leading order in the compactified theory, in the small-L limit, the 2-d field theory can be reduced to a 1-d quantum mechanics, the asymptotic perturbative expansion of the ground state energy in units of the natural frequency ω in CPN −1 model on R × S1 is given by [13, 38], This is a non-alternating series and hence non-Borel summable. The Borel transform of it is, The lateral Borel sums are,
B(g2)θ± =
On the otherhand, the leading ambiguity comes from the bion amplitude,
[Bii]θ± = Re[Bii] + Im[Bii]θ± (4.127) Remarkably, they canel each other exactly,
Im Bθ±+ Im[Bii]θ± = 0, to the order of e−2SIN . (4.129)
This is an evidence that the resurgence relation works in the compactified CPN −1 model with periodic fermions. The exact resurgence relation in CP1 QM has been found by Toshiaki Fujimori,Syo Kamata,Tatsuhiro Misumi,Muneto Nitta and Norisuke Sakai [25, 39], it seems the relation should also work in CPN −1 QM.
5 Conclusion and Future directions
Resurgence theory works well in quantum mechanics. We can define observables in quantum mechanics non-perturbatively by the construction of trans-series. By finding all the non-perturbative saddles in path integral, the physical quantities are well-defined. The coefficients of the expansion still need to be taken care of since we do not know how to express all the coefficients. Some evidences of the non-trivial relation between the coefficients of different saddle points have been found in quantum mechanics [10, 11].
In quantum field theories, because the coupling constant runs with the scale, there are renormalons closer to the origin by a factor of order Ncthen the instantons.
In some theories after compactification, we can find new semiclassical configurations which has the action less then the instantons by a factor of order Nc too. However, only in few theories, the position of the singularity produced by the new semiclas-sical configuration concide with the IR renormalon. In addition, the origin of the renormalons and the instanon-anti-instanton singularities seems to be so different.
One comes from the factorial growth of only one set of Feymann diagrams, another one comes from the factorial growth of various Feymann diagrams. As we have seen in section 2, the renormalon divergence comes from the high momentum and low momentum contribution to the Adler function, which is very different from instanton divergence. We can ask, can renormalon really be realized by semiclassical config-urations, just like instanton-anti-instanton singularities? If it can, then there must be a deep connection between the running of the coupling and those semiclassical configurations. Unfortunately we have not seen this kind of relation yet. Perhaps this kind of semiclassical realization can only be used in quantum mechanics case.
Let’s summarize the problems about the semiclassical realization to renormalons.
The first is why the position of the newly found semiclassical configurations do not
concide with the IR renormalons in some theories. If the position of the renor-malons would change with compactified radius, then how to show that? Second, in some theories, the compactification of one direction to small radius results in phase transition. We cannot continuously change the radius from small to infinity, so even though we find new semiclassical configuration, they cannot correspond to the renormalons in the original theory. Third, as mentioned above, the renoamalons are obviously related to the running coupling, so the new semiclassical configuration should also related to it, but we did not find such relation until now.
Acknowledgement
I offer my sincerest gratitude to my academic advisor Kazuo Hosomichi. Thanks for your encouragement, patience and the assistance. This thesis would never have been if without you. I would also like to thank those who have taught me durung the tume that I have stayed in National Taiwan University: Heng-Yu Chen, Pei-Ming Ho,Yu-Tin Huang. Special thanks to my best friend Chia-Wei Chen and those who have accompanied me during the two years: Chih-Kai Chang, Tsung-Hsuan Tsai, Ta-Yu Chiang, En-Jui Kuo and many others.
Appendices
A Ambiguity of Borel resummation
Consider this factorially divergent series, P (g) =
∞
X
n=0
n!gn (A.1)
The sign of this series is non-alternating so it is not Borel summable.. After Borel transformed, it becomes
BP(g) =
∞
X
n=0
gn= 1
1 − g (A.2)
Now the original series P (g) can be represented by BP(g) P (g) =
Z ∞ 0
dt e−tBP(gt) = Z ∞
0
dt e−t 1
1 − gt (A.3)
-6
t
t = 1g
t
Figure 11: The pole on the Borel complex t plane
It has a pole on the positive real axis. In order to compute the integral, we need to do analytic continuation to avoid the pole. We can choose the path to be C+ or C−.
P (g + i) = Z
C+
dt e−t 1
1 − gt = Re P (g) + i Im P (g), (A.4) P (g − i) =
Z
C−
dt e−t 1
1 − gt = Re P (g) − i Im P (g). (A.5)
-6
t
t = 1g
t
((((((((((
hhhhhhhhhh
C+
C−
Figure 12: The different choice of paths
Thus the integral is ambiguous, it has two possible different values. The discon-tinuity is,
P (g + i) − P (g − i) = 2i Im P (g) = Z
C+−C−
dt e−t 1
1 − gt. (A.6) This is a contour integral around the pole at t = 1g, the value is given by 2πi multiplying the residue,
Z
C+−C−
dt e−t 1
1 − gt = 2πi
g e−1g. (A.7)
Therefore, the ambiguous imaginary part of the series P(g) is ±iπge−1/g.
B Computation of the functional determinant in Quantum Mechanics
Consider an operator M (t1, t2) given by (3.33) with some boundary conditions which depend on the boundary condition of the path integral. The determinant of M is realized as the product over its eigenvalues. Let qnbe the orthonormal eigenfunctions of M .
Z
dt2M (t1, t2)qn(t2) = λnqn(t1). (B.1) We can write more precisely,
[−d2
dt2 − V00(qc(t))]qn(t) = λnqn(t), (B.2) and
Z
dt qn(t)qm(t) = δnm. (B.3) The determinant of M is,
detM =Y
n
λn. (B.4)
In general, the operator which we often want to compute can have zero modes and negative modes,
[−d2
dt2 − V00(qc(t))]q0(t) = 0, (B.5) if q0(t) is not 0, then q0(t) is a zero mode of M .
[−d2
dt2 − V00(qc(t))]qa(t) = λaqa(t), (B.6) if λa< 0 then qa(t) is an negative mode of M .
We first discuss the issue of zero mode. Naively, the zero mode makes the functional determinant become zero. We need to be careful with this. What we want to compute is the integral,
Z
D[qf(t)] exp
−1 2
Z
dt1dt2qf(t1)M (t1, t2)qf(t2)
= (det M )−12. (B.7)
If we expand qf(t) by its normalized eigenmodes qn(t) of M and q0(t) is the zero mode,
qf(t) =X
n≥0
cnqn(t), (B.8)
then the integral become, (det M )−12 =
Z Y
n
dcn
√2πe−12Pn≥0λnc2n =
Z dc0
√2π(det0M )−12, (B.9)
where
det0M = Y
n6=1
λn, (B.10)
is the determinant of which zero mode is removed. The integral of c0 results in infinity. In order to compute this, we need to know what is the zero mode q0(t).
Actually, q0(t) is propotional to ˙qc(t). Recall that qc(t) is a solution of the EOM of the Euclidean action.
¨
qc(t) + V0(qc(t)) = 0, (B.11) doing derivative on t, we get,
d2
dt2q˙c(t) + V00(qc(t)) ˙qc(t) = 0, (B.12) thus ˙qc(t) is a zero mode of M . The relation between q0(t) and ˙qc(t) is,
q0(t) = 1
|| ˙qc||q˙c(t). (B.13) The norm is given by,
|| ˙qc(t)||2 = Z β/2
−β/2
dt ˙qc(t)2, (B.14)
we can use the energy conservation (3.19),
|| ˙qc(t)||2 = Z β/2
−β/2
dt 1
2q˙c(t)2− Z β/2
−β/2
dt V (qc(t)) = Sc, (B.15) which is just the action of the instanton trajectory. Note c0 is the collective param-eter of q0(t), t0 is the collective parameter of ˙qc(t). We can find the relation between c0 and t0. Doing variation to q0 with respect to c0,
q0(t)δc0 = 1
|| ˙qc(t)||q˙c(t)δc0. (B.16)
We can also do variation to ˙qc with respect to t0, q0(t)δc0 = 1
|| ˙qc(t)||q˙c(t)δc0 = ˙qc(t)δt0. (B.17) Then we can find the Jacobian of changing c0 to t0, which is,
J = δc0 δt0
= Sc1/2. (B.18)
At the end, the integral of c0 becomes, Z dc0
√2π = Sc1/2
√2π Z β/2
−β/2
dt0 = βSc1/2
√2π . (B.19)
This is why we need to multiply this factor at Sec 3.1.1., it comes from the zero mode integration. Therefore, the determinant (B.9) becomes,
(det M )−12 = βSc1/2
√2π (det0M )−12. (B.20) We have extracted the zero mode from the determinant.
The situation of the negative modes are simpler. A negative mode would make the determinant of M become negative and (det M )12 would become imaginary.
If there is an imaginary part in the ground state energy, it means the potential is unstable. The vacuum which we do expansion is a false vacuum and it would eventually decay. So if the vancum we choose is stable, we don’t need to worry about the negative mode, it will not appear in the operator M .
Appendix B.A Gelfand-Yaglom method
There are several ways to compute the functional determinant. The most intuitive way is to find all the spectrum of the operator, then doing regularization. However, it is hard to do this usually. There is one useful method to compute the functional determinant in Quantum mechanics without knowing all the spectrum. It is known as the Gelfand-Yaglom theorem [40].
Consider a second order differential equation with some boundary conditions at the interval [−β2,β2]. The eigenfunction equation is given by,
M φ(t) = [−d2
dt2 + V (t)]φ(t) = λφ(t), (B.21)
where M = [dtd22 + V (t)] is the operator which we want to compute the determinant, λ is the eigenvalue. We denote φ(1,2)λ (t) to be the solutions with the initial condition,
φ(1)λ (−β/2) = 1, φ(2)λ (−β/2) = 0, φ˙(1)λ (−β/2) = 0, φ˙(2)λ (−β/2) = 1.
(B.22)
These two solutions do not need to satisfy the boundary condition, since the ODE is second order, we can always find solutions satisfy (B.22). We can use them to construct a matrix,
Every solutions of the eigenfunction equation with arbitrary initial condition can be construct by the matrix Eλ(t),
Now we can write down the most general boundary condition for the eigenvalue problem,
where A and B are matrices which depend on the boundary condition. For example, Drichlet: A =
We can use (B.24) to rewrite (B.25), then we find,
[A + BEλ(β
So if this equation is satisfied for some special value of λ, there is a solution of this eigenvalue problem and this λ is the eigenvalue. The condition on λ is,
det so if we want to compute the determinant of M , we only need to construct the matrix Eλ(t). We do not need to know all the spectrum of M . If there are zero modes in M and we want to remove them, we just write,
det0M = − ∂
∂λdet(M − λ)|λ=0. (B.29)
If there are more than one zero modes, we just need to do more derivatives.
Let’s take the periodic boundary condition as an example. First we note A = 1 and B = −1 then the determinant become,
det
Since the Wronskian is constant, det
The condition of λ become, Tr The two linear independent zero modes are given by,
φ0(t) = A ˙qc(t) + B ˙qc(t)
We have known that ˙qc(t) is one of the zero mode solution of M . The other one can be easily found by assuming the solution is ˙qc(t)a(t). Solve a(t) can give us the result (B.35). We can translate t0 to make ¨qc(−β/2) = 0. Then we can find φ(1,2)λ (t) And we obtain the determinant is,
det M = [2 − φ(1)0 (β
2) − ˙φ(2)0 (β
2)], (B.37)
for periodic boundary condition. For other different boundary conditions, we just need to change the matrices A and B.
When we need to remove the zero mode in det M , we need to compute φ(1,2λ (t) to first λ order. We can use φ(1,20 and the Green’s function to construct it. To first order, it is Let’s use the operator M appear in section (3.1.1) as an example. We want to compute (3.46). Note we need to use anti-periodic boundary condition. The operator is given by,
M (t) = −d2
dt2 + 1 − 3
2 cosh2(t/2), (B.40)
there is one zero mode inside this operator. From (B.28) and (B.29), we find, det0M = − ∂ Recall the instanton solution is,
qc(t) = 1
√g 1
1 + et, q˙c(t) = − et
(1 + et)2, (B.43)
then we can compute (B.42). The first integral is just the action of the instanton.
Sc= 1
6g. (B.44)
The second integral is, Z β/2
−β/2
dt0
˙
qc(t0)2 = g Z β/2
−β/2
dt [(e−t2 + e2t)4] = 2g[3β + 8 sinh(β/2) + sinh(β)], (B.45) so the determinant is
det0M = β 2 + 4
3sinh(β/2) + 1
6sinh(β). (B.46)
We also need to compute the reference determinant, which is given by, M0(t) = −d2
dt2 + 1, (B.47)
where
φ(1)0 (t) = cosh(t + β/2), φ(2)0 (t) = sinh(t + β/2), (B.48) for anti-periodic boundary condition,
det M0 = 2 + 2 cosh(β). (B.49)
Finally, after takes the limit β → ∞, det0M det M0 = 1
12. (B.50)
This is the final result we want to know. Using this method, we can compute the determinant without solving the eigenfunction equation completely. This method can also be generalized to Sturm-Liouville problems. See [23].
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